A173036 a(n) = binomial(n+1, 2) + 13.

13, 14, 16, 19, 23, 28, 34, 41, 49, 58, 68, 79, 91, 104, 118, 133, 149, 166, 184, 203, 223, 244, 266, 289, 313, 338, 364, 391, 419, 448, 478, 509, 541, 574, 608, 643, 679, 716, 754, 793, 833, 874, 916, 959, 1003, 1048, 1094, 1141, 1189, 1238, 1288, 1339, 1391

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Feb 07 2010

Keywords

• nonn
• easy
• less

Comments

Numbers m such that 8*m - 103 is a square. - Bruce J. Nicholson, Jul 26 2017

For n>2, a(n+14) is the number of ways to tile an equilateral triangle of side length 2*n+1 with smaller equilateral triangles of side length n and side length 1. For example, with n=3, there are a(17)=166 ways to tile an equilateral triangle of side length 7 with smaller ones of sides 3 and 1, including the one way with 49 triangles of sides 1 and the seven ways with four triangles of sides 3 (and thirteen triangles of sides 1). - Ahmed ElKhatib and Greg Dresden, Sep 02 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217.

Programs

Magma

[13 + Binomial(n+1,2): n in [0..60]]; // G. C. Greubel, Feb 19 2021

Maple

A173036:= n-> 13 + binomial(n+1,2); seq(A173036(n), n=0..60) # G. C. Greubel, Feb 19 2021

Mathematica

Table[n*(n+1)/2 +13, {n,0,5!}]

PARI

a(n)=n*(n+1)/2 + 13 \\ Charles R Greathouse IV, Jun 16 2017

Sage

[13 + binomial(n+1,2) for n in (0..60)] # G. C. Greubel, Feb 19 2021

Formula

a(n) = A000217(n) + 13.

From G. C. Greubel, Feb 19 2021: (Start)

G.f.: (13 -25*x +13*x^2)/(1-x)^3.

E.g.f.: (26 +2*x +x^2)*exp(x)/2. (End)

Sum_{n>=0} 1/a(n) = 2*Pi*tanh(sqrt(103)*Pi/2)/sqrt(103). - Amiram Eldar, Dec 13 2022

Extensions

Title and offset changed by G. C. Greubel, Feb 19 2021